method to linearize the output from an adc

ABSTRACT

A method is disclosed of compensating the output of an ADC for non-linearity in the response of the ADC. The method comprises converting an analog input signal to uncorrected digital ADC output samples, applying a vector of correction variables to each of a block of uncorrected ADC output samples to provide a block of corrected ADC samples, and iteratively minimizing a measure of the spectral flatness of the block of corrected ADC samples with response to the vector of correction variables.

FIELD OF THE INVENTION

This invention relates to analog-to-digital converters (ADC), and inparticular to methods for linearizing the corrected output from suchADCs.

BACKGROUND OF THE INVENTION

The analog-to-digital converter is a class of electronic component whichis used in a wide variety of applications ranging from instrumentationto communications. The basic function of an ADC is to accept an inputwhich is analog—that is to say, it can have any of a continuous range ofvalues—and provide an output in a digital form—that is to say, theoutput must lie within a representation of a subset of the naturalnumbers.

In order to properly carry out their function, the vast majority of ADCsare required to be as nearly as possible, or at least approximately,linear: the output should be directly proportional to the input.

An example of an ADC which requires high linearity is that for a singlefrequency modulated (FM) radio signal, which requires to be linear overthe range of frequency modulation from the base band frequency. Incomparison, an ADC used in a wide band receiver requires linearity overa far wider range: the wide band receiver will sample the entire 20 MHzFM relevant radio band, such as, for instance, a 20 MHz FM radio band.In addition, such a wide band receiver may be exposed to the “near-farproblem” in that it may receive signals from antennas at widelydifferent distances from the receiver: the ADC should be capable ofdigitising signals from both the distant and nearby antenna accurately.Relative to ADCs for narrowband applications, an ADC for such a wideband receiver application thus not only needs to have a much largerbandwidth, but also requires a much higher “spurious free dynamic range”(SFDR). To achieve this requires a very high degree of linearity.

Non-linearity in an ADC response can arise due to a variety of factors.In particular, deviations in the actual resistance or capacitance valuesof manufactured on-chip components such as resistors or capacitors, fromthe design values, can directly result in non-linearity. Thesedeviations can easily arise due to IC process spread, moreover currentswhich exceed the (small signal) approximations of transistors may alsodegrade the linearity. Examples are non-linear circuit parameters inmulti-level quantizers and digital to analogue converters (DACs).

In general, the non-linearities of an ADC may be classified into staticnon-linearities, which manifest themselves for constant input signals,and dynamic non-linearities that only manifest themselves fortime-varying signals. Histograms of the digital sample values for aknown input signal can characterize ADCs with static non-linearities.However, such tests are inadequate to measure ADCs with dynamic errorslike hysteresis.

Historically, addition of a controlled amount of noise to the ADC input,called dithering, has been a popular technique to reduce thenon-linearity of ADCs. Dithering is mostly used for suppression ofnon-linearities in ideal ADCs.

A known method of carrying out linearization of ADCs is to use one of avariety of spectra. Typically such spectra are generated as Fast FourierTransforms (FFT), but other spectra, such as Discrete Cosine Transforms(DCT) may also be used. For example, Adamo et al in “A//D ConvertersNonlinearity Measurement and Correction by Frequency Analysis andDither,” IEEE Tr. Instrumentation Measurement, Vol. 52, No. 4, August2003 assumes a polynomial non-linearity in the ADC, for which a singlecarrier at the ADC input causes a series of harmonics in the ADC output.Based on this assumption, Adamo et al analytically determined theenergies of these harmonics.

Dynamic element matching (DEM) is a kind of randomization technique thatreduces harmonics and intermodulation products in non-ideal ADCs at theexpense of a higher noise floor.

This invention is particularly concerned with sigma delta (ΣΔ) ADCs. ΣΔmodulation is a variation of delta (Δ) modulation: the basic form of thetwo forms of modulation, as applied to a single stage in an ADC, isshown in FIG. 1. In a Δ modulator, an analogue input signal is quantizedin quantizer 2, typically to produce a single bit of the digital outputsignal, although in some known Δmodulators multiple bits may beproduced. The resulting quantized signal is subtracted from the originalinput signal at the (Σ) adder 1, resulting in a remnant or Δ signalwhich is equal to the original signal less the quantized (bit) output.This Δ signal is fed back to the quantizer, for the next stage of theADC conversion. The subsequent quantizer output (of the Δ signal) isintegrated along with the original quantized output, by means ofintegrator 4, and finally the output is filtered in a low pass filter 5.The ΣΔ variation is based on the realisation that integration is alinear operation: that is to say, the integral of (a+b) is equal to (theintegral of a)+(the integral of b). And thus the integrator 3′ may beincluded in the circuit between the adder (Σ) and the quantizer 2.

It is known, for example from the Adamo paper referenced above, tominimise the energy of harmonics in the ADC output signal correspondingto a sine wave ADC input. It is also known to use iteratively theinverses of polynomials. However, these methods require preciseknowledge of the input frequency which also needs to fall on the grid offast Fourier transform (FFT) frequencies corresponding to the sine waveinput.

There is thus an ongoing requirement for a linearization method whichdoes not suffer to the same extent from these requirements.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a method forlinearization of an ADC which does not suffer, to the same extent, fromthe above disadvantages.

According to the present invention there is provided a method ofcompensating the output of an ADC for non-linearity in the response ofthe ADC, the method comprising converting an analog input signal touncorrected digital ADC output samples applying a vector of correctionfactors to each of a block of uncorrected ADC output samples, to providea block of corrected ADC samples, and iteratively minimizing a measureof the spectral flatness of the block of corrected ADC samples withrespect to the vector of correction variables.

Preferably the analog signal comprises a plurality of frequencies.

Advantageously the ADC is a ΣΔ ADC comprising a quantizer and ADC whichtogether form a loop filter. Preferably the step of iterativelyminimizing the spectral flatness of the block of corrected ADC sampleswith respect to the vector of correction variables comprises: estimatinga mismatch function V of the quantizer and a mismatch function W of theDAC, estimating the impulse response h of the loop filter determining afirst impulse response f and a second impulse response g, wherein f andg follow the follow relationships:

f(k)=δ(k)−Σ_(i=1) ^(P) h(i)f(k−i),

g(k)=Σ_(i=1) ^(P) h(i)f(k−i).

wherein δ is the Kronecker −Δ function, and iteratively subtracting acorrection sample e from the uncorrected ADC output samples, wherein eobeys the relationship

${{e(k)} = {{\sum\limits_{s = 10}^{L - 1}\; {{V\left( {y(k)} \right)}*{f(k)}}} + {{W\left( {y(k)} \right)}*{g(k)}}}},$

wherein * denotes the convolution of two functions.

Preferably, the measure of spectral flatness is the spectral flatness)λ(

) of a spectrum {right arrow over (Z)} according to:

${{\lambda \left( \overset{\rightarrow}{Z} \right)} = \frac{\exp\left( {\frac{1}{N}{\sum\limits_{i}\; {\log \left( {Z_{i}}^{2} \right)}}} \right)}{\frac{1}{N}{\sum\limits_{i}{Z_{i}}^{2}}}},$

Alternatively, the measure of spectral flatness may be given by ageneralised spectral flatness of the spectrum {right arrow over (Z)}(with p<q):

${\lambda_{p,q}\left( \overset{\rightarrow}{Z} \right)} = {\frac{\left( {\frac{1}{N}{\sum\limits_{i}\; {Z_{i}}^{p}}} \right)^{\frac{1}{p}}}{\left( {\frac{1}{N}{\sum\limits_{i}{Z_{i}}^{q}}} \right)^{\frac{1}{q}}}.}$

Preferably the compensation is applied over a frequency range of atleast minus 200 MHz to 200 MHz. Linearization over such a wide range offrequency is particularly difficult using conventional methods.

More preferably still the compensation is applied over a frequency rangeof approximately or exactly 0 Hz to approximately or exactly 20 MHz.Application of the method over a frequency range of this order makes themethod particularly suitable for wideband receiver applications.

These and other aspects of the invention will be apparent from, andelucidated with reference to, the embodiments described hereinafter.

BRIEF DESCRIPTION OF DRAWINGS

Embodiments of the invention will be described, by way of example only,with reference to the drawings, in which

FIG. 1 shows schematically typical Δ and ΣΔ modulators used in ADCstages;

FIG. 2 shows a ΣΔ feedback loop as applied in the present inventions;

FIG. 3 shows a spectrum of a training signal comprising five carriersprior to the linearization operation;

FIG. 4 shows the spectrum of an operational signal comprising fourcarriers prior to linearization according to the method of theinvention;

FIG. 5 shows the spectrum of the training signal of FIG. 3 afterapplication of the method according to the invention; and

FIG. 6 shows the spectrum of the operational signal of FIG. 4 afterapplication of linearization according to the method of the presentinvention.

It should be noted that the Figures are diagrammatic and not drawn toscale. Relative dimensions and proportions of parts of these Figureshave been shown exaggerated or reduced in size, for the sake of clarityand convenience in the drawings. The same reference signs are generallyused to refer to corresponding or similar feature in modified anddifferent embodiments

DETAILED DESCRIPTION OF EMBODIMENTS

The ADC output signal during a training phase has a finite length; thatduring the operational phase has a semi-infinite length. For a block ofsamples of the signal x(k), k=0, 1, . . . , the vector notation {rightarrow over (x)}=(x₀, x₁, . . . , x_(N−1)) will be used. Thecorresponding fast Fourier transform (FFT) spectrum is denoted {rightarrow over (X)}=(X₀, X₁, . . . , X_(N−1)). δ(k) is the Kronecker deltafunction, such that δ(k)=0, except for δ(0)=1.

We assume that the impulse response h(k) of the ΣΔ-loop filter satisfiesh(k)=0 for k=0, −1, . . . due to causality and a minimal one unit delayin the loop. Truncate the estimate of h to at most P non-zero (complex)samples, such that

h(k)=0 for k=P+1, P+2 . . . .

The transfer function of the loop is denoted H=FFT_(N)(h).

FIG. 2 depicts the additive errors due to mismatch in the quantizer andDAC inside a ΣΔ feedback loop.

In FIG. 2 the DAC 21 introduces, negatively, an error d(k); this erroris fed through the integrator off amp 22 to the quantizer 23, at whichan additional noise signal n(k) is introduced.

We now let y(k) represent the integer samples produced by the L-levelquantizer in response to the test signal during the training phase. Thatis to say: y_(k)εS={0, 1, . . . , S−1}. Indicator signals i^((s))(k),sεS, are defined as follows.

i ^((s))(k)=1 if y _(k) =s, and 0 otherwise  (1)

A weighted sum of the indicator signals i^((s))(k) equals y(k),

$\begin{matrix}{{y(k)} = {\sum\limits_{i = 0}^{L - 1}\; {{{si}^{(s)}(k)}.}}} & (2)\end{matrix}$

After point-wise multiplication “o” with e.g. a Blackman window w(k),the FFTs of the indicator signals are computed,

{right arrow over (I)} ^((s)) =FFT _(N)({right arrow over (w)}o{rightarrow over (i)} ^((s))).  (3)

The linearity of the FFT as applied to Equation (2) yields that

$\begin{matrix}{{{FFT}_{N}\left( {\overset{\rightarrow}{w} \cdot \overset{\rightarrow}{y}} \right)} = {\sum\limits_{i = 0}^{L - 1}\; {s{{\overset{\rightarrow}{I}}^{(s)}.}}}} & (4)\end{matrix}$

The quantization noise if the quantizer had been ideal is denoted n(k),see FIG. 2. The mismatch error signal q(k) at the input of an non-idealquantizer is estimated with a function V of the current quantizer outputsample y(k), that is q(k)=V(y(k)). Similarly, the mismatch error signale(k) at the input of a non-ideal DAC is estimated by means of a functionW of the current quantizer output sample y(k), that is d(k)=W(y(k)).Observe, that the spectra of q(k) and d(k) follow from equation (4),

$\begin{matrix}{{E_{q} = {\sum\limits_{i = 0}^{L - 1}{{V(s)}{\overset{\rightarrow}{I}}^{(s)}}}},{E_{d} = {\sum\limits_{i = 0}^{L - 1}{{W(s)}{{\overset{\rightarrow}{I}}^{(s)}.}}}}} & (5)\end{matrix}$

From FIG. 2 it is apparent that q(k) and d(k) are convolved with impulseresponses f(k) and g(k),

f(k)=δ(k)−Σ_(i=1) ^(P) h(i)f(k−i),

g _(k+1)=Σ_(i=1) ^(P) h _(i) f _(k−i).

before they reach the quantizer output of the ΣΔ-loop, respectively.

In the frequency domain, the corresponding transfer functions are

F(f)=1/(H(f)+1),G(f)=H(f)F(f),

respectively. Note that for moderate loop gains, we cannot approximateF(f)≈1, G(f)≈0.

Now, Equation (5) implies that the total ADC output error spectrumequals

$\begin{matrix}{{\overset{\rightarrow}{E} = {\sum\limits_{i = 0}^{L - 1}{\left( {{{V(s)}\overset{\rightarrow}{F}} + {{W(s)}{\overset{\rightarrow}{D} \cdot \overset{\rightarrow}{G}}}} \right) \cdot {\overset{\rightarrow}{I}}^{(s)}}}},} & (7)\end{matrix}$

where D(f) is the Fourier transform of a unit delay function,

D(f)=e ^(j2πf/N) , f=0, 1, . . . , N−1.

The right hand side of the Equation (7) can be interpreted as avector-matrix product of a vector of length L with a L×N-matrix whoserows consist of the vectors I^((s)).

As our sources of nonlinearity are filtered, both static and dynamicnonlinearities can occur. Subtraction of the error spectrum {right arrowover (E)} from the ADC output spectrum {right arrow over (Y)} yields thespectrum of the linearized signal, {right arrow over (Z)}={right arrowover (Y)}−{right arrow over (E)}.

Due to IC process spread, the impulse response h of the loop filtervaries from chip to chip, and, in general, needs to be estimated. Othervariables that need to be estimated during the training phase are thevalues of the functions V and W. They can be viewed as tables with Sentries each. Once V, W, and h are known, f and g follow from Equation(6).

For moderate degrees of nonlinearity of the operational amplifier(opamp) that subtracts the ADC input signal from the DAC output signalin a ΣΔ-loop, computer simulations show that they can be corrected forimplicitly, by letting them impact the estimates of mismatch functionvalues V, W of the quantizer and DAC.

The spectrum flatness (measure) (SF) λ(

) of the spectrum {right arrow over (Z)} is defined as [10]

$\begin{matrix}{{\lambda \left( \overset{\rightarrow}{Z} \right)} = {\frac{\exp\left( {\frac{1}{N}{\sum\limits_{i}\; {\log \left( {Z_{i}}^{2} \right)}}} \right)}{\frac{1}{N}{\sum\limits_{i}{Z_{i}}^{2}}}.}} & (8)\end{matrix}$

In general, 0≦λ≦1, where λ=1 occurs only for signals with a perfectlywhite spectrum. Small λ's correspond to estimates |{circumflex over(Z)}_(i)|², i=0, 1, . . . , N−1 of the power spectral density (PDS) withall energy concentrated in a small number of spectral peaks. When inEquation (8) a small constant ε, ε>0, (e.g. a noise variance) is addedto the PSD, only logarithms of positive values occur. Averaging powerspectra of blocks makes the estimate of the PSD more accurate.

Important for this embodiment of the invention is to iterativelyminimise the spectral flatness SF of a block of corrected ADC outputsamples over the correction variables. It is noted that in order tocarry this out, quantitative knowledge about the ADC input sample is notrequired.

Incremental (or multiplicative) steps in the correction variables yieldnew corrected versions of the block of ADC output samples. The searchfollows the steps that lower the SF-value and discards others. In caseof gradient-descent minimization, the gradient can be evaluatednumerically. Too large steps along the gradient vector increase the SF.

During the operational phase of the linearization method we subtract thecorrection samples

$\begin{matrix}{{{e(k)} = {{\sum\limits_{s = 10}^{L - 1}\; {{V\left( {y(k)} \right)}*{f(k)}}} + {{W\left( {y(k)} \right)}*{g(k)}}}},} & (9)\end{matrix}$

from the ADC output samples y(k) to obtain the corrected samplesz(k)=y(k)−e(k), where “a”(k)*“b”(k) denotes convolution of the functionsk→“a”(k) and k→“b”(k).

In an alternative embodiment of the invention, a generalised spectralflatness measure (GSF) is implemented rather than the SF describedabove.

A Generalized Spectral Flatness Function is a function of a power)spectrum, typically a power spectrum, that takes on its maximal valuefor a spectrally flat input signal, and takes on its minimal value, orvalues close to its minimal value, for input signals that are maximallypeaked. A GSF will have a monotonic behaviour in between these twoextrema. Hence, if a signal's spectrum gets to have either fewer peaksor larger peaks, the GSF decreases as the spectrum has larger deviationfrom a flat spectral shape.

A ratio of two functions, in which, as the spectrum becomes more peaked,the denominator increases relatively stronger than the nominatorprovides a measure of the “flatness” of the spectrum. For instance, aratio (with p<q)

$\begin{matrix}{{\lambda_{p,q}\left( \overset{\rightarrow}{Z} \right)} = {\frac{\left( {\frac{1}{N}{\sum\limits_{i}\; {Z_{i}}^{p}}} \right)^{\frac{1}{p}}}{\left( {\frac{1}{N}{\sum\limits_{i}{Z_{i}}^{q}}} \right)^{\frac{1}{q}}}.}} & (11)\end{matrix}$

is invariant to scaling of the spectrum. Dividing by a mean—that is tosay, setting p=1 in equation 11, such that the numerator denotes themean—and a root mean square value, that is to say, setting q=2 inequation 11, such that the denominator denotes the root mean square,minimizes the number of multiplications and their numerical range.

In order to demonstrate the effectivity of the method, it has beenapplied by means of computer simulations, as follows. Gaussian mismatchvalues were introduced in the quantizer and DACs inside a wide band FMADC. The standard deviations were equal to 5% and 1% of the differencebetween the levels of the 16-level quantizer and DAC, respectively. BothADC stages were simulated. The search procedure also optimized the noisecancellation filter. Moderate third order nonlinearities were introducedin the opamps that subtract the ADC input signals from the DAC outputsignals. Thermal noise was modelled by the addition of Gaussian noise tothe ADC input signals.

The training and operational ADC input signal consisted of five and fourcarrier signals respectively. Given the short time that was simulated(0.3 ms), FM modulation of these carriers by audio signals had not madea difference. For the sake of brevity, the input signals are not shown.FIGS. 3 and 4 show the spectra {right arrow over (Y)} of around 150ksamples of the ADC output signals. FIG. 3 shows the training spectrumincluding 5 carrier signals 31, at 0 dB, along with a noise level 33 ataround −90 dB; the error spectra 32, comprising multiple peaks atbetween −90 dB ad −65 dB, is also clearly apparent. FIG. 4 shows acorresponding operational spectrum for four signal carriers 41 at 0 dB;again, the error peaks 42 are apparent above the noise level 43

The correction parameters were varied so as to minimize the GSF, asgiven by Equation (11) with p=1, q=2, of the training signal shown inFIG. 3. The pertaining corrected signal {right arrow over (Z)} is shownin FIG. 5. The absence of error peaks above the noise level 53 (at about−90 dB) is remarkable, thus providing a 90 dB differential between thecarrier signals 51, and the noise background 53. Using the samecorrection parameters, the operational signal of FIG. 4 was corrected.This yielded the signal of FIG. 6. Again, the absence of absence oferror peaks above the noise level 63 (at about −90 dB) is apparent,providing a 90 dB differential between the carrier signals 61, and thenoise background 63.

The main computational complexity of the method is in the (implicit)vector-matrix product in Equation (7) inside the search loop. Due to theover-sampling in the ADC, the FFT spectra originally have a frequencyrange of −218.4 MHz to +218.4 MHz, whereas for the evaluation of theGSF, only a “payload” frequency range of 0 Hz to 20 MHz is needed.Hence, in this embodiment of the invention, the original width N of thematrix in the vector-matrix product can be drastically reduced.

In summary, then, the method described above according to embodiments ofthe invention comprises minimising the spectral flatness of thecorrected ADC output signal, over a vector of parameters of a postprocessing unit. The method implicitly minimises the energy in spuriouscomponents due to mismatch of multi-level quantizers and DACs andmoderate operational amplifier non-linearities in ΣΔ ADCs.

Although the method has been described above in relation to a ΣΔ ADC, itshould be noted that the invention is not limited thereto. Inparticular, the method according to the invention may be used with otherADCs, including, but not limited to, oversampled ADCs and Nyquist rateconverters.

From reading the present disclosure, other variations and modificationswill be apparent to the skilled person. Such variations andmodifications may involve equivalent and other features which arealready known in the art of ADCs and which may be used instead of, or inaddition to, features already described herein.

Although the appended claims are directed to particular combinations offeatures, it should be understood that the scope of the disclosure ofthe present invention also includes any novel feature or any novelcombination of features disclosed herein either explicitly or implicitlyor any generalisation thereof, whether or not it relates to the sameinvention as presently claimed in any claim and whether or not itmitigates any or all of the same technical problems as does the presentinvention.

Features which are described in the context of separate embodiments mayalso be provided in combination in a single embodiment. Conversely,various features which are, for brevity, described in the context of asingle embodiment, may also be provided separately or in any suitablesub-combination.

The applicant hereby gives notice that new claims may be formulated tosuch features and/or combinations of such features during theprosecution of the present application or of any further applicationderived therefrom.

For the sake of completeness it is also stated that the term“comprising” does not exclude other elements or steps, the term “a” or“an” does not exclude a plurality, a single processor or other unit mayfulfil the functions of several means recited in the claims andreference signs in the claims shall not be construed as limiting thescope of the claims.

1. A method of compensating an output of an ADC for non-linearity in aresponse of the ADC, the method comprising converting an analog inputsignal to uncorrected digital ADC output samples, applying a vector ofcorrection variables to each of a block of the uncorrected ADC outputsamples, to provide a block of corrected ADC samples, and iterativelyminimizing a measure of a spectral flatness of the block of correctedADC samples with respect to the vector of correction variables.
 2. Amethod according to claim 1, wherein the analog input signal comprises aplurality of frequencies.
 3. A method according to claim 1, wherein theADC is a sigma delta ADC comprising a quantizer and a DAC which togetherform a loop filter.
 4. A method according to claim 3, wherein the stepof iteratively minimizing the spectral flatness of the block ofcorrected ADC samples with respect to the vector of correction variablescomprises: estimating a mismatch function V of the quantizer and amismatch function W of the DAC, estimating an impulse response h of theloop filter determining a first impulse response f and a second impulseresponse g, wherein f and g follow the follow relationships:f(k)=δ(k)−Σ_(i=1) ^(P) h(i)f(k−i).g _(k+1)=Σ_(i=1) ^(P) h _(i) f _(k−i).  (6) wherein δ is theKronecker-delta function, and iteratively subtracting a correctionsample e from the uncorrected ADC output samples, wherein e obeys therelationship${{e(k)} = {{\sum\limits_{s = 10}^{L - 1}\; {{V\left( {y(k)} \right)}*{f(k)}}} + {{W\left( {y(k)} \right)}*{g(k)}}}},$wherein * denotes the convolution of two functions.
 5. A methodaccording to claim 4, wherein the measure of spectral flatness is thespectral flatness λ(

) of a spectrum {right arrow over (Z)} according to:${\lambda \left( \overset{\rightarrow}{Z} \right)} = {\frac{\exp\left( {\frac{1}{N}{\sum\limits_{i}\; {\log \left( {Z_{i}}^{2} \right)}}} \right)}{\frac{1}{N}{\sum\limits_{i}{Z_{i}}^{2}}}.}$6. A method according to claim 4, wherein the measure of spectralflatness is given by a generalised spectral flatness of the spectrum{right arrow over (Z)}:${{\lambda_{p,q}\left( \overset{\rightarrow}{Z} \right)} = {\frac{\left( {\frac{1}{N}{\sum\limits_{i}\; {Z_{i}}^{p}}} \right)^{\frac{1}{p}}}{\left( {\frac{1}{N}{\sum\limits_{i}{Z_{i}}^{q}}} \right)^{\frac{1}{q}}}.}},$where p<q.
 7. A method according to claim 4, wherein the compensation isapplied over a frequency range of at least −200 MHz to 200 MHz.
 8. Amethod according to claim 4, wherein the compensation is applied over afrequency range of at least approximately 0 Hz to at leastapproximately.